1-3-8 gives the details you request. Whether a player actually participates either without required equipment or with illegal equipment, when the 'violation' is observed, that player is to be sent off the field and may not participate in the next down, unless his team 'buys' him back into the game with a charged team time out. If the equipment issue is noticed while the ball is alive, no, we don't send him off the field while the ball is alive. We wait until the down is over, then send him off WITHOUT interrupting the game. He may not return to the game until he has all required equipment, and no illegal equipment, and has 'sat out' at least one down (unless his team takes a time out).
If he refuses to leave and correct the problem, then the R can take whatever action he deems appropriate, like charging a time out. If the non-compliance and unsportsmanlike conduct persist, then go with UNS on the player. If he continues his insolence, then hit him with another UNS, and wave goodbye to him as you have him escorted off the field by security.